TDB-TT

# Barycentric Dynamical Time for the HP-41

Overview

1°)  Simplified Formula
2°)  INPOP10
3°)  Barycentric Coordinate Time

-Several time scales are available in astronomical work:
-The Terrestrial Time = TT  which is related to the International Atomic Time TAI  by  TT = TAI + 32.184 seconds.

-The Barycentric Dynamical Time = TDB = Temps Dynamique Barycentrique which is preferable for the motions of planets in the Solar system.
-Due to relativistic effects, there is a difference between TDB & TT that the following programs evaluate.
-This difference remains small: less than 2 milliseconds, at least several millennia around J2000.

1°)  Simplified Formula

-Reference [1] gives a very simple formula:

TDB - TT = 0.001658 sin M + 0.000014 sin 2M    where   M is the Sun's mean anomaly.

-But the program hereunder employs the first terms of the series given in reference [2] or [3],
those whose amplitude > 0.4 second over the time-span [1000,3000]
-The errors should not exceed 1 or 2 microseconds over this interval of time.

Data Registers:   R00 thru R03: temp   ( R00 = number of millennia since 2000/01/01  12h )
Flags: /
Subroutine:    "J0" or "J1" or J2"  ( cf "Julian & Gregorian Calendars for the HP-41" )

 01  LBL "TDB"  02  DEG  03  HR  04  24  05  /  06  X<>Y  07  XEQ "J0"  08  +  09  365250  10  /  11  STO 00         12  359993.729  13  *  14  STO 01   15  151  16  +  17  SIN  18  43  19  *  20  RCL 00  21  *  22  RCL 01  23  ST+ X  24  60  25  +  26  SIN  27  17 28  *  29  -  30  RCL 01  31  63  32  +  33  SIN  34  1022  35  *  36  -  37  RCL 00         38  ST* Y  39  585178  40  *  41  STO 02   42  ST+ X  43  RCL 01  44  3  45  *  46  -  47  27  48  -  49  SIN  50  6  51  *  52  -  53  RCL 01  54  ST- 02 55  RCL 00  56  30349  57  *  58  STO 03         59  -  60  ST+ X  61  48  62  -  63  SIN  64  8  65  *  66  -  67  RCL 02        68  9  69  -  70  COS  71  11  72  *  73  +  74  RCL 01   75  RCL 03  76  ST+ X  77  -  78  29  79  +  80  SIN  81  12 82  *  83  -  84  RCL 02         85  ST+ X  86  17  87  -  88  SIN  89  13  90  *  91  +  92  RCL 00   93  4452671  94  *  95  68  96  -  97  SIN  98  16  99  * 100  + 101  RCL 00  102  202 103  * 104  72 105  + 106  SIN 107  17 108  * 109  - 110  RCL 00        111  12221 112  * 113  STO 02 114  42 115  - 116  SIN 117  23 118  * 119  + 120  RCL 01  121  RCL 02 122  - 123  50 124  + 125  SIN 126  47 127  * 128  - 129  RCL 03  130  25 131  + 132  SIN 133  48 134  * 135  + 136  RCL 01        137  ST+ X 138  6 139  - 140  SIN 141  138 142  * 143  + 144  RCL 01 145  RCL 03  146  - 147  65.7 148  + 149  SIN 150  224 151  * 152  - 153  RCL 01  154  2.964 155  - 156  SIN 157  16567 158  * 159  + 160   E7 161  / 162  END

( 259 bytes / SIZE 004 )

 STACK INPUTS OUTPUTS Y YYYY.MNDD / X HH.MNSS TDB-TT (s)

Example1:      2012/08/29  16h41m37s  ( TT or TDB )

2012.0829    ENTER^
16.4137    XEQ "TDB"  >>>>   TDB-TT =  - 0.0013227 s               ---Execution time =18s---

Example2:      3000/04/04  1h23m45s  ( TT or TDB )

3000.0404   ENTER^
1.2345      R/S        >>>>    TDB-TT = + 0.0015386 s

Note:

-In reference [1], the inputs should be expressed in TDB whereas in refence [2], they should be expressed in TT.
-To the level of accuracy of this simplified formula, the difference is negligible.

2°)  INPOP10

-Like the JPL ephemerides DE4xx, the French ephemerides INPOP ( Integrations Numériques Planétaires de l'Observatoire de Paris )
give the positions and velocities of the bodies of the Solar System under the form of Chebyshev Polynomials.
-But instead of the nutation, the difference TT-TDB is also provided: 12 coefficients for each time-span of 4 days, cf reference [4]

Data Registers:        •  R00 = bbb.eee               ( Registers R00 & Rbb thru Ree are to be initialized before executing "TDB2" )

•  Rbb = JD1   •  Rbb+1 = JD2      •  Rbb+2 = a0    •  Rbb+3 = a1  ...............................   •  Ree = an
Flags: /
Subroutines:      "J0" or "J1" or J2"  ( cf "Julian & Gregorian Calendars for the HP-41" )
"CdT"  ( cf "Orthogonal Polynomials §5-b) or "Ephemerides and Chebyshev Polynomials for the HP-41" )

 01  LBL "TDB2"  02  HR  03  24  04  /  05  X<>Y  06  XEQ "J0"  07  2451544.5  08  +  09  RCL IND 00  10  -  11  ST+ Y  12  ISG 00  13  CLX  14  RCL IND 00  15  LASTX  16  -  17  2  18  /  19  /  20  RCL 00  21  X<>Y  22  1  23  ST+ Z  24  ST- 00  25  -  26  XEQ "CdT"  27  CHS  28  END

( 59 bytes / SIZE ??? )

 STACK INPUTS OUTPUTS Y YYYY.MNDD / X HH.MNSS TDB-TT (s)

Example:       3000/04/04  1h23m45s

-In reference [4], we find a text file that contains the following constants:

2816877.00         1st  Julian Date = 3000/03/31  12h  TDB
2816881.00         2nd Julian Date = 3000/04/04  12h  TDB

-0.15254574201273665E-02
-0.17329122056166975E-04
+0.52937767551143974E-06
+0.27550591219803854E-08
-0.34148361610150929E-09
-0.69921867908556206E-11
+0.85016053321457848E-12
+0.21451895272947141E-14
-0.14234824035252474E-14
+0.17399035530231268E-15
-0.19832610480477758E-17
-0.14385810488591148E-17

-After storing these 14 coefficients in, say R01 to R14  ( control number = 1.014 )

1.014  STO 00

3000.0404   ENTER^
1.2345   XEQ "TDB2"  >>>>    TDB-TT = + 0.001538845852 s

Notes:

-The argument of the INPOP ephemeris is TDB ( you can also choose TCB ), so if the input time is expressed in TT,
we must execute "TDB2" again with Time = 1h32m45s + (TDB-TT) above to get the correct result
-In this example, it yields  TDB-TT = + 0.001538845852 s

-Here, the result is unchanged and it will be almost always the same because the HP-41 works with 10 digits.
-Anyway, one iteration is always sufficient.

3°)  Barycentric Coordinate Time

-Anothe time scale may also be used:  TCB = Temps Coordonné Barycentrique = Barycentric Coordinate Time.
-Whereas  TDB - TT  remains smaller than 0.002 second at least several millenia aroud J2000, the difference  TCB - TDB  is much larger:
-It was almost 0 in 1977 but in 2012 , TCB -TDB is about 17 seconds.

Formula:

TCB = TDB + 1.550519768 E-8 ( JDTCB - 2443144.5003725 ) 86400 + 6.55 E-5 seconds

where  JDTCB  is the Julian Date in the TCB scale.

-The program below uses an iteration to compute  TCB - TDB  for a given date expressed in  TDB

Data Registers:   R00-R01:  temp
Flags: /
Subroutine:     "J0" or "J1" or J2"  ( cf "Julian & Gregorian Calendars for the HP-41" )

 01  LBL "TCB"  02  HR  03  24  04  /  05  X<>Y  06  XEQ "J0"  07  +  08  8400  09  +  10  3725 E-7  11  -  12  STO 00  13  CLST  14  STO 01  15  LBL 01  16  CLX  17  86400  18  /  19  RCL 00  20  +  21  1.33964908 E-3  22  *  23  655 E-7  24  +  25  ENTER^  26  X<> 01  27  X#Y?  28  GTO 01  29  END

( 72 bytes / SIZE 002 )

 STACK INPUTS OUTPUTS Y YYYY.MNDD / X HH.MNSS TCB-TDB (s)

Where the input time is expressed in TDB

Example:       3000/04/04  1h23m45s  TDB

3000.0404   ENTER^
1.2345   XEQ "TCB"   >>>>   500.6752393 seconds

References:

[1]  Robin M. Green - "Spherical Astronomy" - Cambridge University Press - ISBN  0-521-31779-7
[2] "Introduction aux Ephemerides Astronomiques" - EDP Sciences - ISBN 2-86883-298-9  ( in French )
[3]  Fairhead & Bretagnon - "An Analytica Formula for the Time Transformation TB-TT" Astronomy & Astrophysics 229, 240-247 ( 1990 )
[4]  http://www.imcce.fr/inpop
[5]  IAU 2006 Resolution B3